865 research outputs found
LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS IN ARROWHEAD FORM
This paper deals with different approaches for solving linear systems of the first order differential equations with the system matrix in the symmetric arrowhead form.Some needed algebraic properties of the symmetric arrowhead matrix are proposed.We investigate the form of invariant factors of the arrowhead matrix.Also the entries of the adjugate matrix of the characteristic matrix of the arrowhead matrix are considered. Some reductions techniques for linear systems of differential equations with the system matrix in the arrowhead form are presented
A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices
We present the submatrix method, a highly parallelizable method for the
approximate calculation of inverse p-th roots of large sparse symmetric
matrices which are required in different scientific applications. We follow the
idea of Approximate Computing, allowing imprecision in the final result in
order to be able to utilize the sparsity of the input matrix and to allow
massively parallel execution. For an n x n matrix, the proposed algorithm
allows to distribute the calculations over n nodes with only little
communication overhead. The approximate result matrix exhibits the same
sparsity pattern as the input matrix, allowing for efficient reuse of allocated
data structures.
We evaluate the algorithm with respect to the error that it introduces into
calculated results, as well as its performance and scalability. We demonstrate
that the error is relatively limited for well-conditioned matrices and that
results are still valuable for error-resilient applications like
preconditioning even for ill-conditioned matrices. We discuss the execution
time and scaling of the algorithm on a theoretical level and present a
distributed implementation of the algorithm using MPI and OpenMP. We
demonstrate the scalability of this implementation by running it on a
high-performance compute cluster comprised of 1024 CPU cores, showing a speedup
of 665x compared to single-threaded execution
Integrated Nested Laplace Approximations for Large-Scale Spatial-Temporal Bayesian Modeling
Bayesian inference tasks continue to pose a computational challenge. This
especially holds for spatial-temporal modeling where high-dimensional latent
parameter spaces are ubiquitous. The methodology of integrated nested Laplace
approximations (INLA) provides a framework for performing Bayesian inference
applicable to a large subclass of additive Bayesian hierarchical models. In
combination with the stochastic partial differential equations (SPDE) approach
it gives rise to an efficient method for spatial-temporal modeling. In this
work we build on the INLA-SPDE approach, by putting forward a performant
distributed memory variant, INLA-DIST, for large-scale applications. To perform
the arising computational kernel operations, consisting of Cholesky
factorizations, solving linear systems, and selected matrix inversions, we
present two numerical solver options, a sparse CPU-based library and a novel
blocked GPU-accelerated approach which we propose. We leverage the recurring
nonzero block structure in the arising precision (inverse covariance) matrices,
which allows us to employ dense subroutines within a sparse setting. Both
versions of INLA-DIST are highly scalable, capable of performing inference on
models with millions of latent parameters. We demonstrate their accuracy and
performance on synthetic as well as real-world climate dataset applications.Comment: 22 pages, 14 figure
Zerofinding of analytic functions by structured matrix methods
We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV--like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples
Infinite invertible arrowhead matrices and applications
Motivated by current applications of the arrowhead matrices of large order, the infinite invertible arrowhead matrices are considered. A method based on a simple but suitable triangular factorization is proposed for obtaining, in the finite-dimensional case, a decomposition of the inverse in O(n) time. This procedure can be applicable to infinite arrowhead matrices factored properly. Some illustrative examples are given
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